Finite Group Databasebeta

← C2×#5806 #87233

Subgroups — C2×#5806 #87233

Order 384.

1900 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×D4C211yesmaximal, characteristic, complemented
2C22×#237C211yesmaximal, characteristic, complemented
3C2×#953C212yesmaximal, complemented
4C22×#217C212yesmaximal, complemented
5#5806C2116yesmaximal, direct product factor
6C2×#939C212yesmaximal, complemented
7C2×#938C211yesmaximal, characteristic, complemented
8C2×#936C211yesmaximal, characteristic, complemented
9C2×#935C212yesmaximal, complemented
10C2×D4×Dic3C211yesmaximal, characteristic, complemented
11C2×C4×#67C211yesmaximal, characteristic, complemented
12C2×#924C211yesmaximal, characteristic, complemented
13C2×#546—31nomaximal, Sylow
14C25×C3C2212yes—
15C22×C3×D4—28no—
16C22×C3×D4C2212yes—
17C22×C3×D4C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×C3×D4C2218yescomplemented
20C23×C3×C4C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2212yescomplemented
23C22×#67C2212yescomplemented
24C23×Dic3C2212yescomplemented
25C23×Dic3C2212yescomplemented
26C2×#237—28no—
27C2×#237C2218yescomplemented
28C22×C4×D3C2211yescharacteristic, complemented
29C22×Dic6C2211yescharacteristic, complemented
30#953C22116yescomplemented
31C2×#217C2212yes—
32C2×#217C2211yescharacteristic
33C2×#217C22116yescomplemented
34C2×#217C2212yescomplemented
35C2×#217C2212yescomplemented
36C2×#217C2212yescomplemented
37C2×#217C2212yescomplemented
38#939C22116yescomplemented
39#938C2218yescomplemented
40#936C2218yescomplemented
41#935C22116yescomplemented
42D4×Dic3C2218yescomplemented
43C4×#67C2218yescomplemented
44C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
45C2×C4⋊Dic3C2211yescharacteristic, complemented
46#924C2218yescomplemented
47C2×Dic3⋊C4C2212yescomplemented
48C2×Dic3⋊C4C2211yescharacteristic, complemented
49C2×C4×Dic3C2211yescharacteristic, complemented
50C22×#41—31no—
51C23×D4D311yescharacteristic, complemented
52#546—316no—
53C2×#124—31no—
54C2×#123—32no—
55C2×C22⋊Q8—31no—
56C2×#121—31no—
57C2×#121—32no—
58C2×#120—32no—
59C2×C4×D4—31no—
60C2×C4×D4—31no—
61C22×C22⋊C4—32no—
62C24×C3—28no—
63C24×C3—28no—
64C24×C3—44no—
65C24×C3C2312yes—
66C24×C3C2318yes—
67C24×C3C2312yes—
68C24×C3C2311yescharacteristic
69C23×D3C2311yescharacteristic
70C2×C3×D4—28no—
71C2×C3×D4—232no—
72C2×C3×D4—24no—
73C2×C3×D4D418yescomplemented
74C2×C3×D4C2318yes—
75C2×C3×D4C2314yes—
76C2×C3×D4C2314yes—
77C22×C3×C4—24no—
78C22×C3×C4C2311yescharacteristic
79C22×C3×C4C2314yes—
80C22×C3×C4C2311yescharacteristic
81C2×#67—28no—
82C2×#67—24no—
83C2×#67—44no—
84C2×#67C2314yes—
85C2×#67C2318yescomplemented
86C2×#67C2318yescomplemented
87C22×Dic3—28no—
88C22×Dic3—44no—
89C22×Dic3C2311yescharacteristic
90C22×Dic3C2311yescharacteristic
91C22×Dic3C2312yes—
92C22×Dic3C2312yes—
93C22×Dic3C2311yescharacteristic
94C22×Dic3C2318yescomplemented
95C22×Dic3C2318yescomplemented
96#237—232no—
97C2×C4×D3—24no—
98C2×C4×D3C2314yescomplemented
99C2×Dic6—24no—
100C2×Dic6C2314yescomplemented
101#217C2318yes—
102#217C2314yes—
103#217C2318yescomplemented
104#217C2318yescomplemented
105#217C2318yescomplemented
106#217C2318yescomplemented
107(C2×D3)⋊C4C2314yescomplemented
108C4⋊Dic3C2314yescomplemented
109Dic3⋊C4C2318yescomplemented
110Dic3⋊C4C2314yescomplemented
111C4×Dic3C2314yescomplemented
112C25C2×D312yes—
113C2×#41—38no—
114C2×#41—68no—
115C22×Q8—31no—
116C22×D4—28no—
117C22×D4—31no—
118C22×D4—32no—
119C22×D4—32no—
120C22×D4C2×D312yes—
121C22×D4C2×D311yescharacteristic
122C22×D4C2×D311yescharacteristic
123C22×D4C2×D318yescomplemented
124C23×C4—32no—
125C23×C4—32no—
126C23×C4—31no—
127C23×C4C2×D311yescharacteristic
128#124—38no—
129#123—316no—
130C22⋊Q8—38no—
131#121—38no—
132#121—316no—
133#120—316no—
134C4×D4—38no—
135C4×D4—38no—
136C2×C4⋊C4—31no—
137C2×C4⋊C4—31no—
138C2×C4⋊C4—32no—
139C2×C22⋊C4—31no—
140C2×C22⋊C4—32no—
141C2×C22⋊C4—32no—
142C2×C22⋊C4—32no—
143C2×C22⋊C4—316no—
144C2×C22⋊C4—32no—
145C2×C22⋊C4—32no—
146C2×C22⋊C4—31no—
147C2×C42—31no—
148C23×C3—22no—
149C23×C3—24no—
150C23×C3—24no—
151C23×C3—22no—
152C23×C3—216no—
153C23×C3—24no—
154C23×C3—24no—
155C23×C3—216no—
156C23×C3—24no—
157C23×C3—48no—
158C23×C3—432no—
159C23×C3C2×D418yes—
160C23×C3#4114yescomplemented
161C23×C3C2414yes—
162C23×C3C2414yes—
163C23×C3C2412yes—
164C23×C3C2411yescharacteristic
165C22×D3—22no—
166C22×D3—22no—
167C22×D3—22no—
168C22×D3C2412yes—
169C3×D4—216no—
170C3×D4—28no—
171C3×D4C2×D4116yescomplemented
172C2×C3×C4—22no—
173C2×C3×C4—28no—
174C2×C3×C4C2×D414yes—
175C2×C3×C4C2412yes—
176C2×C3×C4C2412yes—
177#67—216no—
178#67—28no—
179#67—48no—
180C2×Dic3—24no—
181C2×Dic3—22no—
182C2×Dic3—216no—
183C2×Dic3—22no—
184C2×Dic3—22no—
185C2×Dic3—24no—
186C2×Dic3—48no—
187C2×Dic3C2412yes—
188C2×Dic3C2412yes—
189C2×Dic3C2414yes—
190C2×Dic3C2414yes—
191C2×Dic3C2412yes—
192C4×D3—28no—
193Dic6—28no—
194C24—28no—
195C24—28no—
196C24—31no—
197C24—44no—
198C24C22×D312yes—
199C24C22×D318yes—
200C24C22×D311yescharacteristic
201C24C22×D312yes—
202#41—632no—
203C2×Q8—34no—
204C2×Q8—64no—
205C2×D4—28no—
206C2×D4—232no—
207C2×D4—24no—
208C2×D4—34no—
209C2×D4—38no—
210C2×D4—38no—
211C2×D4—64no—
212C2×D4—68no—
213C2×D4—124no—
214C2×D4#6718yescomplemented
215C2×D4C22×D318yes—
216C2×D4C22×D314yes—
217C2×D4C22×D314yes—
218C22×C4—24no—
219C22×C4—38no—
220C22×C4—31no—
221C22×C4—31no—
222C22×C4—32no—
223C22×C4—31no—
224C22×C4—32no—
225C22×C4—38no—
226C22×C4—34no—
227C22×C4—68no—
228C22×C4—64no—
229C22×C4—124no—
230C22×C4C22×D314yes—
231C22×C4C22×D311yescharacteristic
232C22×C4C22×D311yescharacteristic
233C4⋊C4—34no—
234C4⋊C4—34no—
235C4⋊C4—38no—
236C22⋊C4—34no—
237C22⋊C4—38no—
238C22⋊C4—38no—
239C22⋊C4—38no—
240C22⋊C4—38no—
241C22⋊C4—38no—
242C22⋊C4—34no—
243C42—34no—
244C22×C3—28no—
245C22×C3—216no—
246C22×C3—22no—
247C22×C3—24no—
248C22×C3—24no—
249C22×C3—22no—
250C22×C3—24no—
251C22×C3—24no—
252C22×C3—24no—
253C22×C3—432no—
254C22×C3—48no—
255C22×C3C22×D418yes—
256C22×C3C22×D412yes—
257C22×C3C2×#4112yes—
258C22×C3C2×#4114yescomplemented
259C22×C3H(5,𝔽2)12yes—
260C22×C3C2511yescharacteristic
261C2×D3—22no—
262C2×D3—22no—
263C2×D3—22no—
264C2×D3—44no—
265C3×C4—22no—
266C3×C4C22×D414yes—
267Dic3—24no—
268Dic3—22no—
269Dic3—22no—
270Dic3—22no—
271Dic3—24no—
272C23—24no—
273C23—24no—
274C23—22no—
275C23—22no—
276C23—24no—
277C23—216no—
278C23—24no—
279C23—216no—
280C23—24no—
281C23—32no—
282C23—48no—
283C23—432no—
284C23—62no—
285C23—62no—
286C23—62no—
287C23#23714yescomplemented
288C23C2×#6718yes—
289C23C23×D314yes—
290C23C23×D312yes—
291C23C23×D314yes—
292C23C23×D311yescharacteristic
293Q8—68no—
294D4—216no—
295D4—28no—
296D4—68no—
297D4—616no—
298D4—128no—
299D4C2×#67116yescomplemented
300C2×C4—22no—
301C2×C4—28no—
302C2×C4—32no—
303C2×C4—32no—
304C2×C4—34no—
305C2×C4—32no—
306C2×C4—34no—
307C2×C4—62no—
308C2×C4—62no—
309C2×C4—64no—
310C2×C4—62no—
311C2×C4—64no—
312C2×C4—616no—
313C2×C4—68no—
314C2×C4—128no—
315C2×C4C2×#6714yes—
316C2×C4C23×D312yes—
317C2×C4C23×D312yes—
318C2×C3—24no—
319C2×C3—28no—
320C2×C3—44no—
321C2×C3#54614yescomplemented
322C2×C3C23×D411yescharacteristic
323C2×C3C22×#4111yescharacteristic
324C2×C3C2×H(5,𝔽2)11yescharacteristic
325D3—42no—
326C22—24no—
327C22—24no—
328C22—22no—
329C22—22no—
330C22—28no—
331C22—216no—
332C22—24no—
333C22—24no—
334C22—24no—
335C22—432no—
336C22—48no—
337C22—62no—
338C22—62no—
339C22—62no—
340C22—124no—
341C22C2×#23712yes—
342C22C2×#23714yescomplemented
343C22#100412yes—
344C22C22×#6718yes—
345C22C22×#6712yes—
346C22C24×D311yescharacteristic
347C4—22no—
348C4—62no—
349C4—62no—
350C4—64no—
351C4—62no—
352C4—64no—
353C4C22×#6714yes—
354C3C2×#54611yescharacteristic, complemented, Sylow
355C2—24no—
356C2—28no—
357C2—44no—
358C2—122no—
359C2#580614yesdirect product factor
360C2C22×#23711yescharacteristic
361C2C2×#100411yescharacteristic
362C2C23×#6711yescharacteristic
subgroupG/H
centerC23C23×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC23×C3×D4C2
socleC23×C3C24
hypercenterC23×D4D3
hypocenterC3C2×#546
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×D4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C23×C3×D4C2semidirect
2. C22×#237C2semidirect
3. C2×#953C2semidirect
4. C22×#217C2semidirect
5. #5806C2direct (N × H)
6. C2×#939C2semidirect
7. C2×#938C2semidirect
8. C2×#936C2semidirect
9. C2×#935C2semidirect
10. C2×D4×Dic3C2semidirect
11. C2×C4×#67C2semidirect
12. C2×#924C2semidirect
19. C22×C3×D4C22semidirect
22. C22×#67C22semidirect
23. C22×#67C22semidirect
24. C23×Dic3C22semidirect
25. C23×Dic3C22semidirect
27. C2×#237C22semidirect
28. C22×C4×D3C22semidirect
29. C22×Dic6C22semidirect
30. #953C22semidirect
33. C2×#217C22semidirect
34. C2×#217C22semidirect
35. C2×#217C22semidirect
36. C2×#217C22semidirect
37. C2×#217C22semidirect
38. #939C22semidirect
39. #938C22semidirect
40. #936C22semidirect
41. #935C22semidirect
42. D4×Dic3C22semidirect
43. C4×#67C22semidirect
44. C2×(C2×D3)⋊C4C22semidirect
45. C2×C4⋊Dic3C22semidirect
46. #924C22semidirect
47. C2×Dic3⋊C4C22semidirect
48. C2×Dic3⋊C4C22semidirect
49. C2×C4×Dic3C22semidirect
51. C23×D4D3semidirect
73. C2×C3×D4D4semidirect
85. C2×#67C23semidirect
86. C2×#67C23semidirect
94. C22×Dic3C23semidirect
95. C22×Dic3C23semidirect
98. C2×C4×D3C23semidirect
100. C2×Dic6C23semidirect
103. #217C23semidirect
104. #217C23semidirect
105. #217C23semidirect
106. #217C23semidirect
107. (C2×D3)⋊C4C23semidirect
108. C4⋊Dic3C23semidirect
109. Dic3⋊C4C23semidirect
110. Dic3⋊C4C23semidirect
111. C4×Dic3C23semidirect
123. C22×D4C2×D3semidirect
160. C23×C3#41semidirect
171. C3×D4C2×D4semidirect
214. C2×D4#67semidirect
258. C22×C3C2×#41semidirect
287. C23#237semidirect
299. D4C2×#67semidirect
321. C2×C3#546semidirect
342. C22C2×#237semidirect
354. C3C2×#546semidirect
359. C2#5806direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C23×C3×D42
C22×#2372
C2×#9532
C22×#2172
#58062
C2×#9392
C2×#9382
C2×#9362
C2×#9352
C2×D4×Dic32
C2×C4×#672
C2×#9242
C2×#5463